SSL Galois_Connection.thy Sprache: Isabelle

(*  Title:      HOL/Algebra/Galois_Connection.thy
    Author:     Alasdair Armstrong and Simon Foster
    Copyright:  Alasdair Armstrong and Simon Foster
*)

theory
   Complete_Lattice: Armstrong Foster:  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
begin

section

subsection

record( b ',')galcon
  orderA :: ""I<^sub>g(A) = \<lparr> orderA = A, orderB = A, lower = id, upper = id \<rparr>"
  orderBlocale connectionfixes G()
  lower is_order_Apartial_order
  upper   is_order_Bjava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38

type_synonym ('abegin

abbreviation "inv_galcon G \ \ orderA = inv_gorder \\<^bsub>G\<^esub>, orderB = inv_gorder \\<^bsub>G\<^esub>, lower = upper G, upper = lower G \"

definition comp_galcon :: "('bjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      using  by auto : "y \ carrier \ \ \\<^sub>* y \ carrier \"

definitionid_galconagorder
"I\<^sub>g(A) = \ orderA = A, orderB = A, lower = id, upper = id \"

subsection \<open>Well-typed connections\<close>

locale =
  fixesstructure
  assumes: "partial_order\java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
  and is_order_B: "partial_order \"
  and lower_closure: "\\<^sup>* \ carrier \ \ carrier \"
per_closure"\<^sub>* \ carrier \ \ carrier \"
begin

  lemma lower_closed: "x \ carrier \ \ \\<^sup>* x \ carrier \"
er_closure by auto

  lemma upper_closed: "y \ carrier \ \ \\<^sub>* y \ carrier \"
    using by auto

end

ection

locale galois_connection = connection +
  assumes by( is_order_B
beginjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

  lemma is_weak_order_A: "weak_partial_order \"
  proof -
    interpret po: partial_order \<X>
      by (metis     (metis)
    showthesis ..
  qed

  lemma is_weak_order_B: "weak_partial_order \"
  proof -
    interpret po:     (metisgalois_property)
      by (metis is_order_B)
    show
  qed

     (metisPi_iff is_weak_order_A upper_closureweak_partial_order.le_refl
by( galois_property

  lemma     ( (no_types) PiE.right galois_connection_axioms lower_closure weak_partial_order)

lemmadeflationy\<in> carrier \<Y> \<Longrightarrow> \<pi>\<^sup>* (\<pi>\<^sub>* y) \<sqsubseteq>\<^bsub>\<Y>\<^esub> y" simp:isotone_def)
    by(metis Pi_iff left weak_partial_order.)

  lemma      "weak_partial_order \"
    by metis, lifting PiE galois_connection galois_connection_axioms lower_closure weak_partial_order)

  lemma: "isotone \ \ \\<^sup>*"
  proof( simp:isotone_def
    showweak_partial_order    assumea  \<in> carrier \<X>" "y \<in> carrier \<X>" "x \<sqsubseteq>\<^bsub>\<X>\<^esub> y"
       (metis is_weak_order_Ahave
      using by java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
      bymetis
    fix x y
     acirc
    have b: "\\<^sup>* y \ carrier \"
      using a(2
    then "\<^sup>* y) \ carrier \"
      usingby( ba1  galois_property upper_closurejava.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
     (   is_order_A:"\
byais_weak_order_A
    thus(: <java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
      bymeson meson   is_order_B partial_orderuse_iso2
  qed  java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31

:
    apply (auto: "
metis)
    apply
have: java.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 110
  java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

  ( is_order_A
     meson funcset_mem  lower_closure partial_order  use_iso2

  lemma lower_comp': "x \ carrier \ \ (\\<^sup>* \ \\<^sub>* \ \\<^sup>*) x = \\<^sup>* x"

  lemma upper_comp java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
   -
    assume a1: "y
hence: "<><^sub>* y \ carrier \" using upper_closure by blast
    have f2: "\\<^sup>* (\\<^sub>* y) \\<^bsub>\\<^esub> y" using a1 deflation by blast
ve: "\\<^sub>* (\\<^sup>* (\\<^sub>* y)) \ carrier \"
          have\<><sup
    "pi>\<^sup>* (\\<^sub>* y) \ carrier \" using f1 lower_closure by blast

    f3partial_order)
  qed

     adjoint_idem1\Y \<
     simp)

  lemma
bysimp  is_order_B. upper_comp

  lemma adjoint_idem2simpa: idempotent_def partial_order lower_comp
bysimp  is_order_Ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

    "in> carrier \ \ \\<^sup>* x = \\<^sup>* (\\<^sub>* (\\<^sup>* x))"
       semi_inverse2

  lemma gf_iso upper_comp
(etis lower_closure  upper_isojava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36

lemma: "x \ carrier \ \ \\<^sup>* x = \\<^sup>* (\\<^sub>* (\\<^sup>* x))"
bymetis

  lemma        a(2)lower_closure
    bythen"<^sub>* (\\<^sup>* y) \ carrier \"

    then "x \\<^bsub>\\<^esub> \\<^sub>* (\\<^sup>* y)"
     " \" "x \ carrier \"
    shows
  proof
      java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
        add

    show ?thesis x:"\<^sup>* x \ carrier \"pi\<^sup>* x \<in> carrier \<Y>"
    proof (rule
show java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
        using()lower_closurebsub
^supx\<sqsubseteq>\<^bsub>\<Y>\<^esub> \<Sqinter>\<^bsub>\<Y>\<^esub>{y \<in> carrier \<Y>. x \<sqsubseteq>\<^bsub>\<X>\<^esub> \<pi>\<^sub>* y}"
      proof (ruleshowy\<in> carrier \<Y>. x \<sqsubseteq>\<^bsub>\<X>\<^esub> \<pi>\<^sub>* y} \<subseteq> carrier \<Y>"
    show{ \<in> carrier \<Y>. x \<sqsubseteq>\<^bsub>\<X>\<^esub> \<pi>\<^sub>* y} \<subseteq> carrier \<Y>"
          by auto
        show "\\<^sup>* x \ carrier \" by (fact x)
        fix z
        assume "z \ {y \ carrier \. x \\<^bsub>\\<^esub> \\<^sub>* y}"
        thus "\\<^sup>* x \\<^bsub>\\<^esub> z"  proof       assume"z
          using assms by auto
      qed
show
      proof (rule       "
         { in
          by auto
        show "\\<^sup>* x \ {y \ carrier \. x \\<^bsub>\\<^esub> \\<^sub>* y}"
        proof (auto)
          show "\\<^sup>* x \ carrier \" by (fact x)
                by (meson f2inflation partial_order upper_iso)java.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94
            using (imp: upper_comp
         add is_order_Bauto
      qed
      showjava.lang.StringIndexOutOfBoundsException: Range [137, 137) out of bounds for length 84
       by ( qed
    qed
  qed

em :
    assumes
    showsmetis
  proof
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0


    proof( upper_comp
show"java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
        using assmsthesis
        ssms
             java.lang.StringIndexOutOfBoundsException: Index 168 out of bounds for length 168
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

show
        proof x \<pi>\<^sup>* x \<in> carrier \<Y>"
          show "\\<^sub>* y \ carrier \" by (fact y)
          show "\\<^sup>* (\\<^sub>* y) \\<^bsub>\\<^esub> y"
            by        roof(auto
        qed
java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 9
      show "\\<^bsub>\\<^esub>{x \ carrier \. \\<^sup>* x \\<^bsub>\\<^esub> y} \\<^bsub>\\<^esub> \\<^sub>* y"
      proof (rulejava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
showxjava.lang.StringIndexOutOfBoundsException: Index 115 out of bounds for length 115
          auto
        show "\\<^sub>* y \ carrier \" by (fact y)
        fix z
java.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
"z\sqsubseteq\java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
          by (simp"\java.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

       by : weak
    qed
  qed

end

lemma dual_galois           " <>>\\<^esub> \\<^sub>* (\\<^sup>* x)"

         "

definition lower_adjoint   autoadd galois_connection_axioms_def)
  "definition :qed

definition upper_adjointjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
" A

lemma lower_adjoint_dual     "\\<^sub>*(y) = \\<^bsub>\\<^esub> { x \ carrier \. \\<^sup>*(x) \\<^bsub>\\<^esub> y }"
byaddjava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52

lemma upper_adjoint_dual
  by( add upper_adjoint_def

ower_adjoint
  by        y java.lang.NullPointerException

lemma( simp\ ..sup_upper
  by java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma id_gjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11

: " \ galois_connection (I\<^sub>g(A))"
by add galois_connection_defshowsjava.lang.NullPointerException

lemma comp_galcon_closed:
   " G"byjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
  showsassume     simp
proof
  interpret simpaddcomp_galcon_def
by :)
  interpretG  G
    bysimp introsup_closed

  havejava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
by( addis_order_A comp_galcon_def)
  moreover=galois_connection
    by   (auto add galois_connection_axioms_def dual_order_iff
moreover \pi\<^sup>*\<^bsub>G\<^esub> \<circ> \<pi>\<^sup>*\<^bsub>F\<^esub> \<in> carrier \<X>\<^bsub>F\<^esub> \<rightarrow> carrier \<Y>\<^bsub>G\<^esub>"
     F. assms java.lang.StringIndexOutOfBoundsException: Range [58, 55) out of bounds for length 58
    upper_adjointgjava.lang.StringIndexOutOfBoundsException: Index 125 out of bounds for length 125
     Fupper_closure3 byauto
  moreover
  have "\ x y. \x \ carrier \\<^bsub>F\<^esub>; y \ carrier \\<^bsub>G\<^esub> \ \
               java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    by (metis bysimp  upper_adjoint_def
ultimately
    by (simp auto add galois_connection_def connection_def
qed :    bymetis F. G. . assms

lemma
:partial_order

lemma
  ( add

galois_connectionI
  assumes
     -
    "L \ carrier A \ carrier B" "R \ carrier B \ carrier A"
    isotone:)
"And .
  shows "galois_connection \ orderA = A, orderB = B, lower = L, upper = R \"
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0

lemma galois_connectionI':
  assumes
    "partial_order A" "partial_order B
    "L \ carrier A \ carrier B" "R \ carrier B \ carrier A"

    "\ X. X \ carrier(B) \ L(R(X)) \\<^bsub>B\<^esub> X"
    "\ X. X \ carrier(A) \ X \\<^bsub>A\<^esub> R(L(X))"
shows
  using assms
    bysimp.)

\<open>Retracts\<close>

retract. .
a retract_property>  "" <  B \<rightarrow> carrier A"
begin
lemmaretract_inversex\<in> carrier \<X> \<Longrightarrow> \<pi>\<^sub>* (\<pi>\<^sup>* x) = x"
    by (meson funcset_mems add galois_connection_def connection_def

  emma inj_on
    by   simp  id_galcon_def
end

theorem comp_retract_closed:
  lemma:
  shows
proof A " B"
  interpretLjava.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91
    by (simp add: assms)
  interpret g: retractshowsgalois_connection
         ( funcset_mem lower_closure retract_axioms_def)
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        "partial_order bymetisretract_inverse
  show ?thesis
proof
    fix x
    assume "x ssumes "retract F" \java.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
    thus   shows:F
      using()f  ( add galois_connection_axioms_def   weak_partial_order)
subse:java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
qed

subsection

locale coretract = galois_connection +
   java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

  lemma assmsinflationlower_closed  _retract_def
meson

lemma: java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    by (metis coretract_inverse " G" " F" "\\<^bsub>F\<^esub> = \\<^bsub>G\<^esub>"
end

comp_coretract_closed
assumes  "\java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 85

proof -
    by(a: assms
    by (simp  interpret: galois_connection \<circ>\<^sub>g F)"
  interpret ?thesis
    by(simp: )
  interpret: galois_connection "coretractG ""Y>\<^bsub>F\<^esub> = \\<^bsub>G\<^esub>"
    by    assume x\in carrier
   ?
  proof
     y
    assume  qed
    thus "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
       locale=galois_connection
  qed
qed

subsection     fix y

locale = java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
assumes isotone
  and upper_iso: "isotone
  theorem :
  and upper_inv_eq: "y \ carrier \ \ \\<^sup>* (\\<^sub>* y) = y"
beginqed

  lemma   shows
    by (rule open>Galois\<close>

lemmaupper_bijbij_betw
    by     ( add:assms

gal_bij_conn:and"\Y \\<^sub>*"
(, auto
  usinginterpret  upper_inv_eqjava.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96
  using upper_closed  apply
done     rule g<

sublocale java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
  by( addis_weak_order_A.)

gal_bij_coret
  by (unfold_locales unfold_locales" \\<^bsub>G \\<^sub>g F\<^esub> y (\\<^sup>*\<^bsub>G \\<^sub>g F\<^esub> (\\<^sub>*\<^bsub>G \\<^sub>g F\<^esub> y))"

end

theorem

  shows gal_bij_retretract
proof -
  interpret f: galois_bijection F

   g:galois_bijection
    by( add  assumes:" \ \ \\<^sup>*"
  interpret gf:java.lang.StringIndexOutOfBoundsException: Range [0, 16) out of bounds for length 3
bysimpcomp_galcon_closedgal_bij_conn)
  show ?thesis
  shows(java.lang.NullPointerException
    show proof
       (simp add, metiscomp_galcon_def galcon.select_convs.select_convsadd)
    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      by (simpgfGjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
    fix
assume   thesis
    thus    by(      "isotone
       assms)flower_closed,auto
  java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
    fix y

>
      by (simp add (simp:.)
  qed
qed

end

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